I think what you heard about "booking fast" may have been in reference to the nursery. Unlike the older kids' clubs, the nursery limits the number of hours you can use it per cruise, and it can fill up.
However, even if you pre-register, you still have to stand in line on embarkation day to sign releases and get the Magic Bands to track the kids and control their ability to leave the club. The lines to do this, either in the terminal or at the clubs on the ship, can get long. It seems (to me, at least) that the lines move quicker and aren't nearly as claustrophobic in the terminal vs. on the ship.
You don't have to register for dinner in the main dining rooms or Cabanas. However, if you want to eat in one of the specialty restaurants, you are best-served in doing so ASAP so you can get the reservation time you want.
Book the shore excursions you want ASAP as well, as some of those can fill up. Check your Navigator as soon as you can and scan it for events that require ticketing - Anna & Elsa, Princess Gathering, Character Breakfast, etc. The Navogator should tell you where to go to get those tickets.